Vector and Matrix Material Matrix = Rectangular array of numbers, complex numbers or functions If r rows & c columns, r*c elements, r*c is order of matrix. A is n * m matrix Square if n = m aij is in row i column j A vector has one row or one column For a square matrix: a11, a22, .. ann form the main diagonal 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Examples and MatLab MATLAB is interactive Matrix Laboratory Package Matrices can be defined, for instance, at the prompt. >> A = [21 52 13; 42 10 1] A= 21 52 13 42 10 1 We will now consider some simple matrix operations 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Matrix Addition R = A + B A and B must have same size. Result R also has same size In R for all elements, rij = aij + bij MATLAB Session >> A = [1 3; 2 7]; % define A, dont output it >> B = [8 1; 4 2]; % ditto B >> R = A+B % evaluate sum, assign to R R = 9 4 6 9 >> R = [1 3; 2 7] + [8 1; 4 2]; % dont need to input A and B! 02/01/2019 SEMMA13 Eng Maths and Stats

Matrix Equality A = B Scalar Multiplication R = k*A A = B if they are of the same size and corresponding elements are same ie aij = bij for all i,j Scalar Multiplication R = k*A Each element in A multiplied by a scalar: rij = k * aij. Note, k * (A + B) = k*A + k*B 02/01/2019 SEMMA13 Eng Maths and Stats

Matrix Multiplication R = A * B Number of columns in A must equal number of rows in B R has number of rows as A and number of columns as B e.g; if A is 2 * 3, B is 3 * 4 , A * B is 2 * 4 matrix. Do element 1 of ith row of A * element 1 of jth column of B Also do for remaining elements of these rows and columns Find the sum of each of these products and store in rij MATLAB: >> R = [2 5; 4 3] * [6; 1]; 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Example If A * B is ok, then B * A may not be possible. If both are possible, A * B  B * A, in general. A * (B * C) = (A * B) * C = A * B * C A * (B + C) = (A * B) + (A * C) (k * A) * B = k * (A * B) = A * (k * B) = (A * B) * k {scalar k} 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Example of Use 16.26 o 36.87 300N T 2 1 Resolving forces in horizontal and vertical directions: cos (16.26) T1 = cos (36.87) T2 sin (16.26) T1 + sin (36.87) T2 = 300 This can be expressed in matrices 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Electronic Example i 1 3 2 18 Ω 10 15 12V By Kirchhoff: 12 = 18 i1 + 10 i2 10 i2 = 15 i3 i1 = i2 + i3 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Matrix Transpose Here the matrix is flipped: rows become columns and vv If R = AT, then rij = aji. If A is size m*n, then AT is size n*m. MATLAB: >> A’ 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats More on Transpose Rules (AT)T=A (A+B)T=AT+BT (A*B)T=BT*AT (kA)T=kAT 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Special Matrices If S scalar, A * S = S * A. A* I = A To convert a scalar, k, to a matrix, multiply scalar by I Note In denotes n*n identity matrix, I3 shown above 02/01/2019 SEMMA13 Eng Maths and Stats

Diagonal Matrices and Equations It is easy to evaluate - clearly x = 4 y = 5 and z = 9 02/01/2019 SEMMA13 Eng Maths and Stats

Triangular Matrices and Equations It is quite easy to evaluate: Clearly z = 2 from the 3rd row. Then, row 2 gives 2y + 4*z = 18 But z known, so 2y = 18-8 so y = 5. Then, row 1 gives x + 3*5 + 2*2 = 21 so x = 2 02/01/2019 SEMMA13 Eng Maths and Stats

Determinants and Inverses Consider the weight suspended by wires problem: One (poor) way is to find inverse or ‘reciprocal’ of A, A-1. It is defined that A-1 * A = A * A-1 = I the identity matrix and A * I = I * A = A If we know A-1, then by pre-multiplying the equation: A-1 * A * T = A-1 * Y thus T = A-1 * Y How to find the inverse - if one exists (1/0 not exist) ? 02/01/2019 SEMMA13 Eng Maths and Stats

Determinants, Adjoint and Inverse One way of finding inverse of square matrix … Note, a matrix has no inverse if its determinant is 0. The determinant of A is written as det(A) or | A | The Adjoint of A is denoted Adj(A) Adj(A) is the transpose of the matrix of cofactors of A. If Cij is the cofactor of aij (defined on next slide), then Adj A, = [Cji] = [Cij]T (cofactors use determinants) 02/01/2019 SEMMA13 Eng Maths and Stats

Cofactors and Determinants Cofactor Cij of matrix A is -1(i+j) * Mi,j where Mi,j is determinant of matrix A without row i and column j The determinant of a single element matrix is that element That of a n*n matrix is the sum of each element in its top row multiplied by that element’s cofactor. | A | = 1 * C11 + 2 * C12 + 3 * C13 Note: determinants are not just used to find inverses. In fact, determinants are more useful than inverses. 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Two by Two Matrices 02/01/2019 SEMMA13 Eng Maths and Stats

Applied to Suspended Mass Recall Sensible to leave in this form, so divide by 0.8 only twice i.e. T1 = 300N and T2 = 360N Check T1*0.96 - T2*0.8 = 288 - 288 = 0 T1*0.28 + T2*0.6 = 84 + 216 = 300  02/01/2019 SEMMA13 Eng Maths and Stats

Three by Three Matrices Ugh! Easiest to remember and calculate by cofactor rule... 02/01/2019 SEMMA13 Eng Maths and Stats

Electronic Circuit Example 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Concluded In MatLab, >> a=[18, 10, 0; 0, -10, 15; -1, 1, 1]; >> v = [12; 0; 0]; > > det(a) ans = -600 >> inv(a)*v ans = 0.5000 0.3000 0.2000 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Some Properties (A-1)-1 = A (AT)-1 = (A-1)T (A * B)-1 = B-1 * A-1 { note change of order } If AT = A-1 then matrix A is an orthogonal matrix. For n*n matrix A: | A | = a11 * C11 + a12 * C12 + … + a1n-1 C1n-1 + a1n C1n = a11 * C11 - a12 * M12 + … - a1n-1 M1n-1 + a1n M1n (n even) or = a11 * C11 - a12 * M12 + … + a1n-1 M1n-1 - a1n M1n (n odd) 02/01/2019 SEMMA13 Eng Maths and Stats

Solve Equations Gaussian Elimination Linear equations in examples, in the general form Ax = b. Important Each row is equivalent to one equation. Do ‘row operations’ on each row, row X := rowX*a + rowY*b Aim : make A part of above triangular (so called echelon form) 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Example Aim to turn 2 in row 2 to 0. Row2 := Row1 * 2 - Row2 * 3 = [0 8 -3 -1] Now turn 1 in bottom row to 0 Row3 := Row2 - Row3 * 8 = [0 0 -19 -57] 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats So Hence, z = -57 / -19 = 3; y = (-1 + 9) / 8 = 1; x = (10 - 4) / 3 = 2. 02/01/2019 SEMMA13 Eng Maths and Stats

Gauss Jordan – for Matrix Inverse To invert A matrix, form matrix [A I], do row operations until A part is unit matrix I, Inverse matrix is in second part Zero A(2,1), A(3,1) Zero A(3,2) 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Continued Zero A(1,3), A(2,3) Zero A(1,2 Now set diagonals of the left half of the matrix to 1: Divide Row 1 by 360, Row 2 by 120 and Row 3 by -15 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats MatLab Session >> aa = [ 3 -1 1; -1 3 4; -1 1 2 ]; >> ai = [ aa, eye(3) ] ai = 3 -1 1 1 0 0 -1 3 4 0 1 0 -1 1 2 0 0 1 >> rref (ai) ans = 1.0 0 0 0.20 0.30 -0.70 0 1.0 0 -0.20 0.70 -1.3 0 0 1.0 0.20 -0.20 0.80 >> a = [ 3 4 0; 2 0 1; 0 1 2 ]; >> b = [ 10; 7; 7 ]; >> a\b % does GaussElim ans = 2.0000 1.0000 3.0000 >> rref ([a b]) % row ops 1 0 0 2 0 1 0 1 0 0 1 3 02/01/2019 SEMMA13 Eng Maths and Stats

How many solutions to equations Consider these graphs with 2 lines of form y – m x = c y x First graph: one value of x and y satisfies both equations, at the intersection of the lines: one solution. Second graph: two lines overlap - infinite solutions. Third graph: two lines are parallel - there is no solution. Any set of linear equations has 0, 1 or ∞ solutions 02/01/2019 SEMMA13 Eng Maths and Stats

Finding 0 solutions – by G.E. e.g. 2 x + y + 3z = 4; x + y + 2z = 0; 2 x + 4 y + 6 z = 8 Zero the first column from rows 2 and 3 Row2 := Row1-2*Row2 Row3 := Row1-Row3 Row3 := 3*Row2 - Row3 Last row means 0 = 16! So, there is no solution. 02/01/2019 SEMMA13 Eng Maths and Stats

Finding Infinite Solutions by G E e.g. 2 x + y + 3z = 4; x + y + 2z = 0; 2 x + 4 y + 6 z = -8 Eliminating the first column Row2 := Row1-2*Row2 Row3 := Row1-Row3 Now do Row3 := 3*Row2 - Row3 Last row means 0 = 0; True for all values of x, y and z. ∞ solutions to equations. They are linearly dependent. If there are solutions, equations are linearly independent. 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Rank of Matrix Can determine how many solutions using Rank of a matrix Rank of matrix is largest square submatrix whose det ≠ 0. A submatrix of A is A minus some rows or columns It has four 3*3 submatrices. All whose det = 0 But, for instance So Rank is 2 02/01/2019 SEMMA13 Eng Maths and Stats

Fundamental Theorem of Linear Systems If system defined by m row matrix equation A x = b The system has solutions only if Rank (A) = Rank (Ã ) If Rank (A) = m, one solution If Rank (A) < m, infinite number of solutions Properties of Rank The Rank of A is 0 only if A is the zero matrix. Rank (A) = Rank (AT) Elementary row operations don't affect matrix rank. Rank is a concept quite useful in control theory. This leads to two related and useful topics…… 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Homogeneous Systems If system is A x = 0, i.e. b = 0, system is homogenous. Such a system has a trivial solution, x1 = x2 = .. xn = 0. A non trivial solution exists if Rank(A) < m. Cramer's Rule For homogeneous systems If D = | A |  0, the only solution is x = 0 If D = 0, the system has ‘non trivial’ solutions This is useful for eigenvalues and eigenvectors. And then there is ……. 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Cramer’s Theorem Alternative to Gaussian Elimination Solutions to a linear system A * x = b , (A is an n*n) x1 = D1/D x2 = D2/D .... xn = Dn/D where D is det|A| D <> 0 and Dk is det of matrix formed by taking A and replacing its kth column with b. Impracticable in large matrices : as hard to find det. 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Example Suspended Mass Here, D = 0.96 * 0.6 - -0.8 * 0.28 = 0.8 Replacing first column of A with b, determinant is : Thus T1 = 240 / 0.8 = 300 N Replacing 2nd column of A with b, determinant is: Thus T2 = 288 / 0.8 = 360 N 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Circuit Example D = |A| found earlier as –600 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Other Functions MatLab has commands such as A^3 rank(a) trace(a) exp(a) 02/01/2019 SEMMA13 Eng Maths and Stats

Eigenvalues and Eigenvectors Consider the equation A x =  x Here A is n*n matrix, x an n*1 vector and  a scalar For a given A we want to find  and thence x The n scalars  satisfying A x =  x are the eigenvalues of A (or characteristic value or latent root of A) For each , an x satisfying A x =  x is an eigenvector of A Sometimes eigenvalues are labelled as  Equation can be reorganised as A * x – λ * x = 0 or (A – λ * I) x = 0 Homogeneous equation, trivial solution x = 0. By Cramer’s Rule, non trivial solution if |A - λ*I| = 0 n eigenvalues found by solving characteristic equation |A - λ*I| = 0 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats 2*2 Example |A - λ*I| = (-4 - λ)(-3 - λ) - 2*1 = λ2 + 4λ + 3λ +12-2 = λ2 + 7λ + 10 In this case, the characteristic equation is λ2 + 7λ + 10 = 0 Characteristic polynomial is λ2 + 7λ +10 = (λ + 5) (λ + 2) The solutions for the characteristic equation are λ = -5 and λ = -2 These are the two eigenvalues for the matrix A. SEMMA13 Eng Maths and Stats 02/01/2019

Finding Eigenvectors of A Eigenvectors of A are x satisfying (A - λ*I) x = 0 for each λ This represents equations: x1 + x2 = 0 and 2*x1 + 2*x2 = 0 Not independent - infinitely many solutions - always happens So choose x1=1, say, then x2=-1. Thus an eigenvector is As are all multiples of this vector (recall Ax = λx) 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Other Eigenvector This represents : -2*x1 + x2 = 0 and 2*x1 - x2 = 0 Again not independent: let x1=1, so x2=2. So eigenvector 02/01/2019 SEMMA13 Eng Maths and Stats

Application - Diagonalisation To diagonalise an n*n matrix A. Let its eigenvalues and eigenvectors are λ1, λ2, .. λn, and Λ1, Λ2, .. Λn. Let X = [Λ1 Λ2 .. Λn] , then |A- λI| = (5 - λ)(2 - λ) - 4 = 0 or λ2 - 7λ + 6 = 0 Eigenvalues are 1 and 6. 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats MatLab Session >> a = [-4 1; 2 -3]; >> [evec eval] = eig(a) evec = % Note, MATLAB arranges that -0.7071 -0.4472 % eigenvector is normalised 0.7071 -0.8944 % (sum of squares of eval = % eigenvector values set to 1) -5 0 % eigenvalues down diagonal 0 -2 >> inv(evec)*a*evec % to show diagonalisation -5.0000 0 0.0000 -2.0000 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Singular Values For an n*n matrix A. Form the symmetric matrix ATA Find the eigenvalues of this matrix, λ1, λ2, .. λn The singular values are λr These are used in Principal Component Analysis, Singular Value Decomposition, etc 02/01/2019 SEMMA13 Eng Maths and Stats

Positive Definite Matrices Suppose A is a symmetric n x n matrix, A = AT. For any non zero column vector x, containing n real numbers A is Positive Definite if xTAx is positive (> 0) It is Positive Semi Definite if xTAx  0 A is positive definite if all its eigenvalues are positive. A symmetric matrix is positive definite if: all the diagonal entries are positive, and each diagonal entry is greater than the sum of the absolute values of all other entries in the corresponding row/column. If xTAx is negative, A is negative definite (its eigenvalues all < 0) 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Vectors A scalar has magnitude: e.g. mass, speed, temperature, position A vector has magnitude and direction: e.g. force, velocity Can define position in (x,y,z) terms, e.g. P = (1, 2, 5), Q = (4, -2, 1) A vector defines movement from one position to another Vector P to Q has components 4-1 in x, -2-2 in y and 1-5 in z In general, if P is point (x1, y1, z1) and Q is (x2, y2, z2) then the vector from P to Q, is (x2-x1, y2-y1, z2-z1) 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Some Notation A vector is a row/column matrix: eg for 3D v = [x y z] For 2 dimensions, a vector has 2 components v = [x y] Can extend concept to n dimensions v = [d1 d2 .. dn] The magnitude (or length) of vector v is given by n dimensional Pythagoras e.g. v = [3 4] |v| = (9 + 16) = 5 v = [3 4 12] |v| = (9 + 16 + 144) = 13 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Norm’s of vectors This is the vector length, sometimes called the Euclidean distance This is another norm, sometimes called Taxicab or Manhattan norm This is the Lp norm (p≥1 and real) If p = 1, is Manhattan norm If p = 2, is Euclidean norm 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Matrix Norm For n*m matrix A and m*1 vector x, the p-norm of A is Here sup is the supremum or least upper bound For p is 1, the norm is the maximum of the sum of each column in A For p is 2, the norm is the largest singular value of A 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Vectors in 2D P(1,2) Q(4,-2) y x P to Q = (4-1, -2-2) = (3, –4) Q to P = (1-4, 2—2) = (-3, 4) = -(P to Q) P to Q then Q to R equiv to P to R Suggests a + b = c P = (px, py) Q = (qx, qy) R = (rx, ry) a = [qx-px qy-py], b = [rx-qx ry-qy] c = [rx-px ry-py] a + b = [qx-px + rx-qx qy-py + ry-qy ] = [rx-px ry-py] = c P Q y x R a b c 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Orthogonal vectors A convenient representation is to use i,j (and k) These have length 1 and are at right angles to each other i = [1 0] or [1 0 0] j = [0 1] or [0 1 0] k = [0 0 1] Clearly unit length: |i| = |j| = |k| = 1 j i y x In use: v = [x y] = xi + yj e.g. v = [2 3] = 2i + 3j In 3D: v = [x y z] = xi + yj + zk 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Dot products The dot or inner product of two vectors a and b is a scalar: a.b = |a||b| cos (θ) where θ is angle between a and b a b θ For 2 orthogonal vectors ( = 90O), their dot product is 0: i . j = 0 a . b = (a1i + a2j) . (b1i + b2j) = a1i . b1i + a1i . b2 j + a2 j . b1i + a2 j . b2 j = a1*b1 + a2*b2 e.g. [2 3] . [5 7] = 2*5 + 3*7 = 31 In general, if a = [ a1 a2 ... an ] and b = [ b1 b2 ... bn ] then a.b = a1*b1 + a2*b2 + ... + an*bn 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Vector Cross Product Vector cross product of 2 vectors is also a vector: v = a x b v is a vector that is at right angles to both a and b Length of v is defined as |a| |b| sin (θ) where θ is angle between a and b The components of v are [a2b3 - a3b2 a3b1 - a1b3 a1b2 - a2b1] Tricky to remember – but there is a trick from Matrices a b θ v 02/01/2019 SEMMA13 Eng Maths and Stats

Vector Products and Determinants i x j = k j x k = i k x i = j j x i = -k k x j = -i k x i = -j Scalar Triple Product Geometric applications exist for these products 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Examples Suppose Force F applied to object moving it a distance d  is angle between direction of movement and of F Work done = (length of d) * (component of F in direction d) This is: |d| * |F| cos () = F.d e.g. F = [3 4], d = [6 0]. Work done = 3*6 + 4*0 = 18 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Another Example q P Q r f Force f applied to object at Q. What is moment of force on object about P? If force f is applied at distance d, & f  to d, moment is f * d Required force is |f| sin (); distance at which it acts is |r| Therefore the moment is |r||f| sin () = r  f e.g. P is at (3,2,4), Q is at (7,3,6) and f = [4 3 1] So r = 4i + 1j + 2k and f = 4i + 3j + 1k 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Back Face Elimination surface 2 vectors on surface vector normal to surface Eye 'normal' vector Angle ~120 O Thus surface visible In Computer Graphics, object has surfaces Display surfaces of objects which face eye – find by Is vector normal (perpendicular) to surface pointing towards eye? vs1, vs2 are vectors on surface vn = vs1 x vs2 is normal to surface ve is vector from eye to the surface Surface visible if angle between ve and vn < -90O or > 90O Then cos(angle) < 0, so ve.vn < 0 NB ve.vn = ve.(vs1 x vs2) – i.e. a triple scalar product 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Vector Space and Sets A linear vector space is a set V whose elements are vectors on whom addition and multiplication can act. So if vectors x and y are in V : x, y  V, and a and b are real numbers: x + y  V and x + y = y + x a zero vector 0  V exists such that for all x  V , x + 0  V for x  V, a * x  V for x,y  V, a (x + y) = ax + ay; (a + b)x = ax + bx These properties apply to n , n dimensions of real numbers R A subset U of n , denoted U  n is open if for all vectors x  U a neighbourhood, y, of x can be found, within a distance . A set is closed if its complement in n is an open set 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats More on Sets A point x is a boundary point of a subset U  n if every neighbourhood of x contains at least one point belonging to U and one not. The boundary of U is the set of all boundary points of U, denoted ∂U Point x is a boundary point of a subset U  n if every neighbourhood of x contains at least one point belonging to U and one not. The boundary of U is the set of all boundary points of U, denoted ∂U The interior of a set A is A - ∂A The closure of set, Ā is the union of A and its boundary An open set is equal to its interior, a closed set is equal to its closure A subset U  n is bounded if there is a r > 0 such that A set S  n is compact if it is closed and bounded 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats SEMMA13 Complex Numbers Consider solutions of quadratic equation x2 + 4x + 13 = 0 In general complex number z, has the form a + j b a is the ‘real’ part a = Re(z) b is the ‘imaginary’ part b = Im(z) In the above example have -2 + j3 and -2 –j3 The latter is the ‘complex conjugate’ of the former If z = a + jb its complex conjugate denoted z is a - jb. 02/01/2019 SEMMA13 Eng Maths and Stats

Rules on Complex Numbers z = a + jb and w = c + jd are equal only if a = c and b = d. z + w = (a + c) + j(b + d) z - w = (a - c) + j(b - d) z * w = (a + jb)(c + jd)= ac + ajd + jbc + j2bd But j2 = -1 so that z * w = ac + ajd + jbc - bd= ac - bd + j(ad + bc) z * z = (a + jb)(a - jb) = a2 - a(jb) + (jb)a - j2b2 = a2 - j2b2 = a2 + b2 This is always a real number ≥ 0 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats On Division So we multiply ‘top and bottom’ by the complex conjugate of the denominator, by c - jd (this is called rationalising): In many practical applications of complex numbers, we don’t need to do this, as we shall see 02/01/2019 SEMMA13 Eng Maths and Stats

Graphical Representation Normal numbers – represented by point on infinite line -∞ to +∞ 5 -1 2 3 4 -2 1 Complex numbers – represent on infinite 2D plane – Argand plane horiz axis for real part, vertical axis for imaginary Re(z) Im(z) b z = a + j b a This is Cartesian representation This leads to an alternative way of representing complex numbers 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Polar Representation Use r, instead of a,b – for same complex number Re(z) Im(z) b z = a + j b  r a r, modulus, |z| is distance from origin , argument, z, is angle from real axis Complex Number can be represented by a and b or r and  Can interchange between Cartesian and Polar forms 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Adding on Argand Plane Re(z) Im(z) b z = a + j b a Re(z) Im(z) d w = c + j d c Re(z) Im(z) b+d z = a + j b a+c w = c + j d a + j b + c + j d = a+c + j(b+d) Adding easy in Cartesian form, more difficult in Polar form 02/01/2019 SEMMA13 Eng Maths and Stats

Multiplying and Dividing If want modulus and argument, easier to multiply in polar form Division also easier : don’t need to use conjugate etc 02/01/2019 SEMMA13 Eng Maths and Stats

Multiplying in Polar Form Re(z) Im(z) b z = r  a  Re(z) Im(z) d w = s  c  Re(z) Im(z) z = r  r*s +  + Multiplying easy in Polar form 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Exponential Form It can be shown, see later on Taylors series, that Exponential Form useful in various integrals 02/01/2019 SEMMA13 Eng Maths and Stats

Hyperbolic and Trigonometric Fn ejθ = cos θ + j sin θ and e-jθ = cos(-θ) + j sin (-θ), add/sub gives: But So 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats De-Moivre’s Theorem (ejθ)p = ejpθ suggesting that (cos θ + jsin θ)p = (cos pθ + jsin pθ) Fine where p is an integer, but when p is a fraction be careful As cos θ = cos (θ+2π) = cos (θ+2kπ) for all integer k So ejθ = ej(θ+2kπ) 02/01/2019 SEMMA13 Eng Maths and Stats

SEMMA13 Eng Maths and Stats Complex Matrices Matrices can have complex numbers as elements As such can have conjugate matrix The eigenvalues of a Hermitian matrix are real; Those of a skew-hermitian matrix are imaginary or zero If eigenvalues of a matrix are complex (conjugate pairs), so are its eigenvectors 02/01/2019 SEMMA13 Eng Maths and Stats